http://hdl.handle.net/11716/57 Mon, 06 Jul 2026 14:06:25 GMT 2026-07-06T14:06:25Z http://hdl.handle.net/11716/14066 Fibonacci polynominals of order $k$ Górowski, Jan; Łomnicki, Adam In this paper we give formulas determining the Fibonacci polynomials of order $k$ using the so-called generalized Newton symbols, i.e., the coefficients in the expansion of $(1+z +z^2 +. . .+z^{k−1})^n$ with respect to the powers of $z$. Wed, 01 Jan 2014 00:00:00 GMT http://hdl.handle.net/11716/14066 2014-01-01T00:00:00Z http://hdl.handle.net/11716/14061 Calculus without the concept of limit Błaszczyk, Piotr; Major, Joanna There are two different approaches to nonstandard analysis: semantic (model-theoretic) and syntactic (axiomatic). Both of these approaches require some knowledge of mathematical logic. We present a method based on an ultrapower construction which does not require any mathematical logic prerequisites. On the one hand, it is a complementary course to a standard calculus course. On the other hand, since it relies on a different intuitive background, it provides an alternative approach. While in standard analysis an intuition of being close is represented by the notion of limit, in nonstandard analysis it finds its expression in the relation is infinitely close. As a result, while standard courses focus on the ε − δ technique, we explore an algebra of infinitesimals. In this paper, we offer a proof of the theorem on the equivalency of limits and infinitesimals, showing that calculus can be developed without the concept of limit. Wed, 01 Jan 2014 00:00:00 GMT http://hdl.handle.net/11716/14061 2014-01-01T00:00:00Z http://hdl.handle.net/11716/14060 Nowe cechy równoboczności trójkątów Biesiada, Michał; Górowski, Jan; Łomnicki, Adam In this paper the authors formulate and prove several conditions for triangle to be equilateral. These conditions are associated with Gergonne, Nagel and Torricelli points and were obtained by composing and solving the so-called ‘enforcement tasks’. Both, the method and the results, can be used to trigger some mathematical student activities at different levels of education or even some teacher activities. Wed, 01 Jan 2014 00:00:00 GMT http://hdl.handle.net/11716/14060 2014-01-01T00:00:00Z http://hdl.handle.net/11716/13660 Approximation by some combinations of the Poisson integrals for Hermite and Laguerre expansions Krech, Grażyna; Wachnicki, Eugeniusz The aim of this paper is the study of a rate of convergence of some combinations of the Poisson integrals for Hermite and Laguerre expansions. We are able to achieve faster convergence for our modified operators over the Poisson integrals. We prove also the Voronovskaya type theorem for these new operators. Tue, 01 Jan 2013 00:00:00 GMT http://hdl.handle.net/11716/13660 2013-01-01T00:00:00Z